As a consequence of a celebrated result of Riesz [1]}, \(P_+\)  is bounded from \(L^q (\)  to itself for every finite \(q > 1\) , a fact that was quantified by Hollenbeck and Verbitsky [2]} when they proved the following sharp inequality:
\(\Vert  P_+ G \Vert _{H^q} \le \csc \left( \dfrac{\pi }{q} \right) \Vert  G \Vert _{L^q}, \quad \forall G \in L^{q} (.\) 
